Conditioning for Variance Reduction in Estimating the Sensitivity of Simulations

We consider first a discrete event static system that is to be simulated at values of a parameter or vector of parameters θ. The system is assumed driven by an input X where typically X is a vector of variables whose density f θ(x) depends on the parameter θ. For the purpose of optimizing, finding roots, or graphing the expected performance E θ L(X) for performance measure L, it is useful to estimate not only the expected value but its gradient. An unbiased estimator for the latter is the score function estimator $$L(X)S(\theta )=L(X)\frac{\delta} {\delta \theta} ln f(\theta )(x)$$ This estimator and likelihood ratio analogues typically requires variance reduction, and we consider conditioning on the value of the score function for this purpose. The efficiency gains due to performing the Monte Carlo conditionally can be very large. Extension to discrete event dynamic systems such as the M/G/1 queue and other more complicated systems is considered.